One would expect that the supremum in (REF ) actually reaches the dimension, which is indeed the case when \(X=\Sigma _a\)  and \(p=1\) : By [1]} we have the variational principle that
\(\underline{\dim }_H A_{\mathbf {I}}= \sup _{\mu \in \mathcal {M}_e(\Sigma _a)} \frac{h_\mu -h_{\mu ,\mathbf {I}}}{-\log \delta },\) 
